WitCH 5: What a West

This one’s shooting a smelly fish in a barrel, almost a POSWW. Sometimes, however, it’s easier for a tired blogger to let the readers do the shooting. (For those interested in more substantial fish, WitCH 2, WitCH 3 and Tweel’s Mathematical Puzzle still require attention.)

Our latest WitCH comes courtesy of two nameless (but maybe not unknown) Western troublemakers. Earlier this year we got stuck into Western Australia’s 2017 Mathematics Applications exam. This year, it’s the SCSA‘s Mathematical Methods exam (not online. Update: now online here and here.) that wins the idiocy prize. The whole exam is predictably awful, but Question 15 is the real winner:

The population of mosquitos, P (in thousands), in an artificial lake in a housing estate is measured at the beginning of the year. The population after t months is given by the function,.

The rate of growth of the population is initially increasing. It then slows to be momentarily stationary in mid-winter (at t = 6), then continues to increase again in the last half of the year.

Determine the values of a and b.

Go to it.

Update

As Number 8 and Steve R hinted at and as Damo nailed, the central idiocy concerns the expression “the rate of population growth”, which means P'(t) and which then makes the problem unsolvable as written. Specifically:

In the second paragraph, “it” has a stationary point of inflection when t = 6, which is impossible if “it” refers to the quadratic P'(t).

On the other hand, if “it” refers to P(t) then solving gives a < 0. That implies P”(0) = 2a < 0, which means “the rate of population growth” (i.e. P’) is initially decreasing, contradicting the first claim of the second paragraph.

The most generous interpretation is that the examiners intended for the population P, not the rate P’, to be initially increasing. Other interpretations are less generous.

No matter the intent, the question is inexcusable. It is also worth noting that even if corrected the question is awful, a trivial inflection problem dressed up with idiotic modelling:

Further Update

The marking schemes for the exam are now up, here and here. As was predicted, “the rate of growth of the population” was intended to mean “population”. As is predictable, the grading scheme gives no indication that the question is garbled garbage.

The gutless contempt with which certain educational authorities repeatedly treat students and teachers is a wonder to behold.

This means that the second derivative would need to be positive. But the second derivative is 6t+2a. Which means that 2a would have to be positive, which means that it is impossible for the rate of growth to be stationary when t=6.

On a side note, I have learned to be suspicious of the wording “find the values of a and b” here in Victoria. In other parts of the world the question would be worded “find the value of a and the value of b” to avoid the question of whether there are multiple acceptable values of a and/or b that the examiner is later going to report that 99% of students missed (whereas actually 1449/1450 is not the same as 99%)

Perhaps using a modified version of Gompertz Law (1825) would be a more sensible approach to modelling mosquito populations as there is an asymptotic limit when all the resources of the lake has been exhausted.

Thanks, Steve. I don’t know a thing about mathematical modelling, and had to click on your link to find out what a Gompertz function is. Real modelling is a difficult art, but of course the “modelling” in Methods is invariably garbage.